Nuclear multilinear operators with respect to a partition (Q1928214)

In one of his theorems, Grothendieck showed that a bounded linear operator \(T:X \to L_1(\mu)\) is nuclear if and only if there exists \(f \in L_1(\mu, X^*)\) such that \(T(x)(\cdot)=f(\cdot)x\). This result can be found in the book [\textit{A. Pietsch}, Eigenvalues and \(s\)-numbers. Cambridge University Press (Orig. Leipzig: Akademische Verlagsgesellschaft Geest \& Portig K.-G.) (1987; Zbl 0615.47019)]. In this relevant paper, the author defines and explores the necessary apparatus to extend this theorem to the multilinear context, namely, the concept of \(n\)-linear nuclear operator with respect to a partition \(\pi\), whose class is denoted by \(\mathcal{N}^\pi\). It is proved that this new class extends the concept of nuclear multilinear operators and is a Banach ideal with an interesting minimal property. The author achieves this goal by proving that a bounded \(n\)-linear operator \(T:X_1 \times \cdots \times X_n \rightarrow L_1(\mu,Y)\) is nuclear with respect to a partition \(\pi\) if and only if there exists \(f \in L_1(\mu,\mathcal{N}^\pi(X_1,\dots,X_n;Y))\) such that \(T(x_1,\dots,x_n)(\cdot)=f(\cdot)(x_1,\dots,x_n)\).